Problem: Chepi is an ecologist who studies the change in the narwhal population of the Arctic ocean over time. She observed that the population loses $5.6\%$ of its size every $2.8$ months. The population of narwhals can be modeled by a function, $N$, which depends on the amount of time, $t$ (in months). When Chepi began the study, she observed that there were $89{,}000$ narwhals in the Arctic ocean. Write a function that models the population of the narwhals $t$ months since the beginning of Chepi's study. $N(t) = $
Answer: The strategy We can model the situation with an exponential function of the general form A ⋅ B f ( t ) A\cdot B\^{ f(t)}, where $A$ is the initial quantity, $B$ is a factor by which the quantity is multiplied over constant time intervals, and $f(t)$ is an expression in terms of $t$ that determines those time intervals. Let's use the given information to determine $A$, $B$, and $f(t)$. Understanding what's given We are given that the initial number of narwhals is $89{,}000$, and that the population loses $5.6\%$ of its size every $2.8$ months. Note that losing $5.6\%$ is the same as being multiplied by $0.944$. [Why?] This means that the initial quantity is $A=89{,}000$ and the factor is $B=0.944$. We need to find $f(t)$ based on the fact that the quantity is multiplied by $0.944$ every $2.8$ months. Finding the expression in the exponent We know that the population of narwhals is multiplied by $0.944$ every $2.8$ months. This means that each time $t$ increases by $2.8$, $f(t)$ increases by $1$. Therefore, $f(t)$ is a linear function whose slope is $\dfrac{1}{2.8}$. When the initial measurement is made, the population hasn't changed. So $N(0) = 89{,}000$, which means that $f(0)=0$. [Why?] Therefore, $f(t)$ must be $\dfrac{t}{2.8}$. Summary We found that the following function models the population of narwhals $t$ months since the beginning of Chepi's study. N ( t ) = 89,000 ⋅ ( 0.944 ) t 2.8 N(t)=89{,}000\cdot (0.944)\^{ \frac{t}{2.8}}